COEFFICIENTS OF DRINFELD MODULAR FORMS AND HECKE …armana.perso.math.cnrs.fr/armana-coefficientsdmfhecke-jnt.pdf · COEFFICIENTS OF DRINFELD MODULAR FORMS AND HECKE OPERATORS 3 Moreover,b

COEFFICIENTS OF DRINFELD MODULAR FORMS ANDHECKE OPERATORS

by

Cécile Armana

Abstract. — Consider the space of Drinfeld modular forms of fixed weight and type forΓ0(n) ⊂ GL2(Fq[T ]). It has a linear form bn, given by the coefficient of tm+n(q−1) in thepower series expansion of a type m modular form at the cusp infinity, with respect to theuniformizer t. It also has an action of a Hecke algebra. Our aim is to study the Heckemodule spanned by b1. We give elements in the Hecke annihilator of b1. Some of themare expected to be nontrivial and such a phenomenon does not occur for classical modularforms. Moreover, we show that the Hecke module considered is spanned by coefficients bn,where n runs through an infinite set of integers. As a consequence, for any Drinfeld Heckeeigenform, we can compute explicitly certain coefficients in terms of the eigenvalues. Wegive an application to coefficients of the Drinfeld Hecke eigenform h.

1. Introduction

Drinfeld modular forms are certain analogues over Fq[T ] of classical modular forms,introduced by D. Goss [12, 13]. A Drinfeld modular form f has a power series expansionwith respect to a canonical uniformizer t at the cusp infinity. If f has type m, thisexpansion is

∑n≥0 bn(f)tm+n(q−1). On the space of Drinfeld modular forms of fixed

weight and type, we have the linear form bn : f 7→ bn(f) and an action of a Hecke algebra.In the present work, we investigate the Hecke module spanned by b1.

Our interest in the problem comes from the torsion of rank-2 Drinfeld modules. In aprevious work, we established a uniform bound on the torsion under an assumption onthe latter Hecke module in weight 2 and type 1 (see [1, 2]). This condition was requiredfor studying a Drinfeld modular curve at a neighborhood of the cusp infinity, namely forshowing that the map from the curve (or rather a symmetric power) to a quotient of itsJacobian variety is a formal immersion at this cusp in a special fiber.

Before stating the main results, we fix some notations. Let A = Fq[T ] be the ring ofpolynomials over a finite field Fq in an indeterminate T , K = Fq(T ) the field of rationalfunctions, K∞ = Fq((1/T )) and C∞ the completion of an algebraic closure of K∞. Foran ideal n of A, k ∈ N and 0 ≤ m < q− 1, we consider the C∞-vector space Mk,m(Γ0(n))of Drinfeld modular forms of weight k and type m for the congruence subgroup Γ0(n) ofGL2(A) (see Section 4.1 for the definition). These are rigid analytic C∞-valued functions

2 C. ARMANA

on C∞−K∞ which have an interpretation as multi-differentials on the Drinfeld modularcurve attached to Γ0(n).

Let T = Tk,m(Γ0(n)) be the Hecke algebra, that is the commutative subring ofEndC∞(Mk,m(Γ0(n))) spanned over C∞ by all Hecke operators TP for P monic polynomialin A (see Section 4.2). Its restriction T′ = T′k,m(Γ0(n)) to the subspace M2

k,m(Γ0(n)) ofdoubly cuspidal forms (with expansion vanishing at order ≥ 2 at all cusps) stabilizes thissubspace. As Goss first observed, doubly cuspidal Drinfeld modular forms play a rolesimilar to classical cusp forms.

In this work, we are interested in the pairing between the space M2k,m(Γ0(n)) and

the Hecke algebra T′ given by the coefficient b1 of the expansion. More precisely,the dual space HomC∞(Mk,m(Γ0(n)),C∞) has a natural right action of T (given bycomposition) and contains the linear form bn : f 7→ bn(f). Let u = uk,m,n : T′ →HomC∞(M2

k,m(Γ0(n)),C∞) be C∞-linear map defined by s 7→ b1s. Our main resultsconcern the kernel I and the image b1T′ of u.

Let Ad+ be the set of monic polynomials of degree d in A. The first statement gives afamily of elements of I.

Theorem 1.1. — The following elements of T′ belong to I:1.∑P∈A1+ P

1−mTP + T1 if m ∈ 0, 1.2.∑P∈Ad+

Ci0P,0 · · ·Cid−1P,d−1TP if d ≥ 1 and (i0, . . . , id−1) ∈ Nd is such that

0 ≤ ij ≤ q −m for all j ∈ 0, . . . , d− 1(1)i0 + . . .+ id−1 ≤ (d− 1)(q − 1)−m.(2)

Here, CP,j ∈ A stands for the jth coefficient of the Carlitz module at P (see Section3.1 for its definition).

3.∑P∈Ad+

P lTP if 0 ≤ l ≤ q −m and d ≥ 1 + (l +m)/(q − 1)∑P∈Ad+

TP if d ≥ 2, or if d = 1 and m = 0.

These elements actually belong to the span over A of all Hecke operators. Moreover,they are universal in the sense that, for a given type m, they do not depend on the weightk nor on the ideal n.

In most cases, we believe that I 6= 0, that is at least one element of Theorem 1.1 isa nontrivial endomorphism of M2

k,m(Γ0(n)), hence the pairing is not perfect. Over thespace M2

2,1(Γ0(n)) with n prime, the situation is as follows. If n has degree 3, we provethat I = 0 (Theorem 7.7). If n has degree ≥ 5, numerical experiments suggest that I 6= 0(Conjecture 6.9). Moreover, it may happen that some elements of Theorem 1.1 are zeroin T′2,1(Γ0(n)): examples of such a situation are explored in Section 6.3.

For the rest of the introduction, we restrict our attention to Drinfeld modular formsof type 0 or 1. Our second statement gives an infinite family of coefficients of Drinfeldmodular forms in b1T′.

Theorem 1.2. — Assume q is a prime and m ∈ 0, 1. Let S be the set of naturalintegers of the form c/(q − 1), where c ∈ N is such that the sum of its base q digits isq − 1. For every n ∈ S , there exists sn ∈ T′, independent of k and n, satisfying

bn = b1sn ∈ b1T′.

COEFFICIENTS OF DRINFELD MODULAR FORMS AND HECKE OPERATORS 3

Moreover, b1T′ is the C∞-vector space spanned by bn for all n ∈ S .

The primality assumption on q is not essential (see Remark 7.3). As for the set S , itis infinite of natural density zero and the first integer not belonging to S is q + 1. Forexample, if q = 3, the first elements of S are

1, 2, 3, 5, 6, 9, 14, 15, 18, 27, 41, 42, 45, 54, 81.Theorem 1.2 relies on an explicit version, Theorem 7.2 (the elements sn that we producedepend on whether the type is 0 or 1). The expression for sn is rather natural: it is aA-linear combination of Hecke operators TP , with P of fixed degree, involving Carlitzbinomial coefficients in A.

Suppose now that I 6= 0. Then the map u fails to be surjective (see Lemma 6.2). Inparticular, b1T′ does not contain all linear forms bn for n ≥ 1. It is then natural to askwhat is the smallest integer n such that bn /∈ b1T′. Theorem 1.2 suggests that n = q + 1might be a good candidate.

Both theorems bring new insight on Drinfeld Hecke eigenforms. Consider a Drinfeldmodular form f which is an eigenform for the Hecke algebra T. Theorem 1.1 translatesinto linear relations among eigenvalues of f , provided that bn(f) 6= 0 for some n ∈ S(Proposition 6.5 and Corollary 7.5). Similarly, Theorem 7.2 gives explicit formulas forcoefficients bn(f) (n ∈ S ) in terms of eigenvalues of f and b1(f). From Theorem 7.2, wealso derive:

– multiplicity one statements in some spaces of Drinfeld modular forms of smalldimension (Theorem 7.7); as far as we know, these are the only known results ofthis kind for Drinfeld modular forms.

– explicit expressions for some coefficients of the Drinfeld modular form h (Proposition8.1). This extends previous work of Gekeler.

As a side remark, we give a brief account of the multiplicity one problem for Drinfeldmodular forms. Since there exist two Hecke eigenforms for GL2(A) with different weightsand same system of eigenvalues (Goss [12]), the question of multiplicity one should bestated as: do eigenvalues and weight determine the Hecke eigenform, up to a multiplicativeconstant? (see Gekeler [7], Section 7). Böckle and Pink showed that this does not holdfor doubly cuspidal forms of weight 5 for the group Γ1(T ) when q > 2 by means ofcohomological techniques (Example 15.4 of [3]). Except for Theorem 7.7 mentionedabove, the question remains open for Γ0(n).

We now compare our results with their analogues for classical modular forms. Considerthe space Sk(Γ0(N)) of cuspidal modular forms of weight k for the subgroup Γ0(N) ⊂SL2(Z) (N ≥ 1). Let (cn(f))n≥1 be the Fourier coefficients of such a modular form fat the cusp infinity. Computing the action of the nth Hecke operator Tn on the Fourierexpansion of f gives the well-known relation, for any n ≥ 1(3) cn(f) = c1(Tnf).In particular, the Hecke module spanned by the linear form c1, which now contains allcoefficients cn, is the whole dual space of Sk(Γ0(N)) and the coefficient c1 gives riseto a perfect pairing over C between Sk(Γ0(N)) and the Hecke algebra. Conjecture 6.9and Theorem 1.2 thus suggest a phenomenon specific to the function field setting. ForDrinfeld modular forms, the reason for not having straightforward statements about the

4 C. ARMANA

kernel and image of u is that the action of Hecke operators on the expansion is not wellunderstood. Goss [12, 13, 11] and subsequently Gekeler [7] wrote down this actionusing Goss polynomials. But such polynomials are difficult to handle (see also Remark5.3). In particular, a relation as general as (3) is lacking.

We now sketch the proofs of Theorems 1.1 and 1.2, which involve rather elementarytechniques.

– We first compute the coefficient b1(TP f), for any f and P , using Goss polynomials(Proposition 5.5). Note that the formula we get is more intricate than (3): it isa A-linear combination of several coefficients of f . For the next step, the crucialpoint is that the index of these coefficients depends only on the degree of P . Thisalready proves that b1T′ is contained in the C∞-vector space spanned by bn, forn ∈ S when m ∈ 0, 1 (Corollary 5.8).

– We take advantage of characteristic p. For power sums of polynomials of a givendegree in A, vanishing properties and closed formulas are well-known (see [21, III]for a survey). Here we use a variant consisting of power sums of coefficients of theCarlitz module. Such sums are studied in Section 3 and closed formulas are givenin Proposition 3.5. In Section 3.4, we also explain their connection with Carlitzbinomial coefficients and special values of Goss zeta function at negative integers.

– By taking adequate linear combinations of b1(TP f), for P of fixed degree, and usingresults of Section 3, we obtain elements in the kernel I (Theorem 1.1, Section 6)and in the image b1T′ (Theorems 7.2 and 1.2).

For the study of the Hecke module b1T′, our method has reached its limit and improvingour results would require new ideas. Our approach might be used to tackle other Heckemodules biT′: however, computing bi(TP f) for any i ≥ 2 is a harder combinatorialproblem.

2. Notations

A tuple will always be a tuple of nonnegative integers. For such a tuple i = (i0, . . . , is),let(i0+...+is

i

)be the generalized multinomial coefficient (i0+...+is)!

i0!···is! .Let q be a power of a prime p and Fq (resp. Fp) be a finite field with q (resp. p)

elements. We will use repeatedly the following theorem of Lucas:(i0+...+is

i

)is nonzero in

Fp if and only if there is no carry over base p in the sum i0 + . . .+ is.We keep the same notations as in the introduction. On A = Fq[T ], we have the usual

degree deg with the convention deg 0 = −∞. By convention, any ideal of A that we willconsider is nonzero. We will often identify an ideal p of A with the monic polynomialP ∈ A generating p. Accordingly, deg p stands for degP .

Let K∞ = Fq((1/T )) be the completion of K at 1/T with the natural nonarchimedeanabsolute value | · | such that |T | = q. We write C∞ for the completion of an algebraicclosure of K∞: it is an algebraically closed complete field for the canonical extension of| · | to C∞.

For P,Q in A, (P ) denotes the principal ideal generated by P , P | Q means P dividesQ and (P,Q) is the g.c.d. of P and Q. The integer part is denoted by b·c.


3. Power sums of Carlitz coefficients

3.1. The Carlitz module. — Let Aτ the noncommutative ring of polynomials inthe indeterminate τ with coefficients in A for the multiplication given by τa = aqτ(a ∈ A). By the map τ 7→ Xq, the ring Aτ can be identified with the subring ofEndC∞(Ga) of additive polynomials of the form

∑aiX

qi (where the multiplication law isgiven by composition). The Carlitz module is the rank-1 Drinfeld module C : A→ Aτdefined by CT = Tτ0 + τ . For a ∈ A, we put Ca for the image of a by C, as usual, andCa =

∑deg ak=0 Ca,kτ

k with Ca,k ∈ A. In particular, Ca,0 = a and Ca,d = 1 if a is monic ofdegree d.

3.2. Deformation of the Carlitz module. — We study the dependence of Ca,k inthe coefficients of a, when a is viewed as a polynomial in T . For this purpose, we need aformal version of the Carlitz module. Let Fq[T,a] = Fq[T,a0,a1, . . .] be the polynomialring in T and an infinite set of indeterminates aii≥0. Consider the ring homomorphism

C : Fq[T,a] −→ Fq[T,a]τ

defined byCT = Tτ0 + τ, Cai = aiτ0 for all i ≥ 0

where the noncommutative ring Fq[T,a]τ is defined in the obvious way. Let P be anelement of Fq[T,a] and d its degree as a polynomial in T . We define CP,0, . . . ,CP,d inFq[T,a] by CP =

∑di=0 CP,iτ

i. These coefficients satisfy the following recursive formulas.

Lemma 3.1. — Let P ∈ Fq[T,a] be a monic of degree d in T . Write P = Tb+ c, withc ∈ Fq[a] and b ∈ Fq[T,a] monic of degree d− 1 in T . Then

CP,0 = TCb,0 + c = P

CP,i = TCb,i + Cqb,i−1 (1 ≤ i ≤ d− 1)

CP,d = Cqb,d−1 = 1.

Proof. — Since C is additive, we have CP,i = CTb,i + Cc,i. Moreover, Cc,i is c if i = 0and 0 otherwise. It remains to compute CTb,i in terms of Cb,i. We have the followingequalities in Fq[T,a]τ:

CTb = CTCb = (Tτ0 + τ)(d−1∑i=0

Cb,iτi

)= T

(d−1∑i=0

Cb,iτi

)+d−1∑i=0

Cqb,iτ

i+1.

By identification, we get our claim.

Lemma 3.2. — Let d ≥ 1 and P ∈ Fq[T,a] monic of degree d in T . Write P =T d + nd−1T

d−1 + . . .+ n0 with n0, . . . , nd−1 ∈ Fq[a]. For all 0 ≤ j ≤ d− 1, one has

CP,j = nqj

j + TQj with Qj ∈ Fq[T, nk | k > j].

In particular, if P = T d + ad−1Td−1 + . . .+ a0, the polynomial CP,j is independent of

a0 for j ≥ 1.

6 C. ARMANA

Proof. — For j = 0, we have CP,0 = P = n0 + T (n1 + . . .+ nd−1Td−1) which has the

expected form. For other coefficients, we proceed by induction on d. The statementis already proven for d = 1. Suppose the property satisfied for all monic polynomialsof degree < d in T . Let P = T d + nd−1T

d−1 + . . . + n0 and write P = Tb + n0 withb ∈ Fq[T, n1, . . . , nd−1] monic of degree < d in T . Let 1 ≤ j ≤ d− 1. By Lemma 3.1, wehave

(4) CP,j = TCb,j + Cqb,j−1.

By hypothesis, there exists Rj−1 ∈ Fq[T, nk | k > j] and Rj ∈ Fq[T, nk | k > j + 1]such that Cb,j = nq

j

j+1 + TRj and Cb,j−1 = nqj−1

j + TRj−1. Substituting in (4), weget CP,j = nq

j

j + T (nqj

j+1 + TRj + T q−1Rqj−1). Since nqj

j+1 + TRj + T q−1Rqj−1 belongsto Fq[T, nk | k > j], the coefficient CP,j has the expected form. The property is thenestablished for any monic polynomial P of degree d.

3.3. Power sums of Carlitz coefficients. —

Notation 3.3. — Let d ≥ 1. Recall that the set of monic polynomials of degree d in Ais denoted by Ad+. For P ∈ Ad+ and i = (i0, . . . , id−1), let

C(P )i = Ci0P,0 · · ·CidP,d = Ci0P,0 · · ·C

id−1P,d−1

(the last equality follows from CP,d = 1). By convention, 00 = 1. Let

Sd(i0, . . . , id−1) =∑

P∈Ad+

C(P )i ∈ A.

Note that for d = 1, the sum is just S1(i) =∑P∈A1+ P

i. We will compute Sd(i0, . . . , id−1)for small i0, . . . , id−1.

Lemma 3.4. — Let 0 ≤ i ≤ 2(q − 1) and P ∈ A. Then

∑a∈Fq

(P + a)i =−1 if i = q − 1 or 2(q − 1)0 otherwise.

Proof. — The vanishing case is merely an application of Lemma 3.1 of Goss [10]. Sincewe need to compute the remaining cases, we give a full proof. Let Ri(P ) =

∑a∈Fq(P +a)i.

Then by the binomial formula,

Ri(P ) =i∑

k=0

(i

k

)P i−k(

∑a∈Fq

ak).

Recall that∑a∈Fq a

k equals −1 if k > 0 and k ≡ 0 mod (q − 1), and 0 otherwise. ThusRq−1(P ) = −1 and Ri(P ) = 0 if 0 ≤ i < q − 1. Now let i = q + j with 0 ≤ j ≤ q − 2.Then

Ri(P ) =∑a∈Fq

(P q + a)(P + a)j = P qRj(P ) +∑a∈Fq

a(P + a)j .


Since j ≤ q − 2, Rj(P ) is zero. Moreover, by the binomial formula,

∑a∈Fq

a(P + a)j =j∑

k=0

(j

k

)P j−k(

∑a∈Fq

ak+1)

which is 0 if j < q − 2 (resp. −1 if j = q − 2).

Proposition 3.5. — Let ij ∈ 0, . . . , 2(q − 1) for all j ∈ 0, . . . , d− 1. Then

Sd(i0, . . . , id−1) =

(−1)d if, for all j, ij ∈ q − 1, 2(q − 1)0 otherwise.

Proof. — The sum Sd(i0, . . . , id−1) is equal to∑a0,...,ad−1∈Fq

Ci0T d+ad−1T d−1+···+a0,0 · · ·C

id−1T d+ad−1T d−1+···+a0,d−1.

By Lemma 3.2, the polynomials CT d+···+a0,1, . . . , CT d+···+a0,d−1 do not depend on a0, sowe can rewrite the sum as∑

a1,...,ad−1∈FqCi1T d+···+a1T,1 · · ·C

id−1T d+···+a1T,d−1

∑a0∈Fq

(T d + . . .+ a1T + a0)i0 .

Let εj be −1 if ij ∈ q− 1, 2(q− 1) and 0 otherwise. Since 0 ≤ i0 ≤ 2(q− 1), Lemma 3.4gives

∑a0∈Fq(T

d + . . .+ a1T + a0)i0 = ε0. Then, again by Lemma 3.2, Sd(i0, . . . , id−1) isequal to

ε0∑

a2,...,ad−1∈FqCi2T d+···+a2T 2,2 · · ·C

id−1T d+···+a2T 2,d−1

∑a1∈Fq

(TQ1 + aq1)i1

.Since 0 ≤ i1 ≤ 2(q−1), Lemma 3.4 yields

∑a1∈Fq(TQ1+aq1)i1 =

∑a1∈Fq(TQ1+a1)i1 = ε1.

Continuing in this fashion, we obtain Sd(i0, , . . . , id−1) = ε0 · · · εd−1.

3.4. Connection with Carlitz binomial coefficients and special zeta values.— We recall Carlitz’s analogue

ak

in Fq[T ] of the binomial coefficient

(nk

)(the reader

may consult Thakur’s article [21] for examples of such analogies). Let a ∈ A and k ∈ Nwith base q expansion

∑wi=0 kiq

i (0 ≤ ki < q). We put ak

=∏wi=0C

kia,i (if i > deg a,

Ca,i = 0 by convention). In particular, aqi

= Ca,i. Note that if 0 ≤ ij < q, then

C(P )i = Ci0P,0 . . . Cid−1P,d−1 =

P

i0 + i1 + . . .+ id−1qd−1

.

In general (ij ≥ q), it is still possible to write Ci0P,0 . . . Cid−1P,d−1 in terms of several Carlitz

binomials. We now explain how Proposition 3.5 might be proved using this formalism.If x is an indeterminate,

xk

is a polynomial in K[x] with degree k (because

xk

is also the exponential function of a finite lattice, see Equation 2.5 of [21] or [14]).Any polynomial f in K[x] may therefore be written as a linear combination of

xk

.

Moreover, the coefficients of this combination can be recovered, in terms of xk

, by a

Mahler inversion type formula due to Carlitz (Theorem 6 in [4], Lemma 3.2.14 in [14]or Theorem XIV in [21]). For f = 1, the coefficients in the binomial basis are easily

8 C. ARMANA

computable and, by the inversion, we obtain for d ≥ 0 and 0 ≤ i < qd with base qexpansion

∑d−1j=0 ijq

j ,

Sd(i0, . . . , id−1) =∑

P∈Ad+

Pi

=

(−1)d if i = qd − 10 otherwise.

This is precisely a special case of Proposition 3.5 (see also [21] p. 14 and Theorem 3.2.16in [14] for similar statements). It seems likely that Proposition 3.5 can be proved byMahler inversion.

Finally, we explain how, by the previous observations, Sd(i0, . . . , id−1) is related tospecial zeta values of Goss zeta function at negative integers. Consider the Carlitz zetafunction ζ : N→ K∞ defined by ζ(k) =

∑P∈A,P monic P

−k. In [10] Goss proved that ζcan be extended to Z by summing over fixed degree: ζ(−k) =

∑∞i=0(

∑P∈Ai+ P

k) ∈ A fork ≥ 0. Now, let p be a prime ideal of A and Ap the ring of integers of the completionof K at p. Following Thakur [21], one can attach to ζ an Ap-valued zeta measure µdetermined by its kth moment:∫

Ap

xkdµ =ζ(−k) if k > 00 if k = 0.

By Wagner’s Mahler-inversion formula for continous functions on Ap ([14] or TheoremVI in [21]), the measure µ is uniquely determined by the coefficients of its divided powerseries i.e. the sequence µk =

∫Ap

xk

dµ (k ≥ 0). Thakur has computed explicitly µk ([21],

Theorem VII). It follows from his proof that, when 0 ≤ ij < q and i = i0 + . . .+ id−1qd−1,

Sd(i0, . . . , id−1) = µi+qd .

4. Drinfeld modular forms and Hecke operators

We collect some basic facts, and set up notation and terminology as well, for Drinfeldmodular forms and Hecke operators.

4.1. Drinfeld modular forms. — The first occurrence of Drinfeld modular formsgoes back to the seminal work of D. Goss [12, 13]. Subsequent developments in the1980s are due to Gekeler [5, 7].

The so-called Drinfeld upper-half plane is Ω = C∞ −K∞, which has a rigid analyticstructure. For an ideal n of A, the Hecke congruence subgroup Γ0(n) is the subgroupof matrices

(a bc d

)∈ GL2(A) such that c ∈ n. Fix an integer k ≥ 0 and a class m in

Z/(q − 1)Z. From now on, m will denote the unique representative of such a class in0, 1, . . . , q − 2. A Drinfeld modular form (for Γ0(n)) of weight k and type m is a rigidholomorphic function f : Ω→ C∞ such that

(5) f

(az + b

cz + d

)= (ad− bc)−m(cz + d)kf(z) for any

(a bc d

)∈ Γ0(n)

and f is holomorphic at the cusps of Γ0(n). We will not detail the second assumptionand rather refer to [5] (V, Section 3) and [15] (Section 2). For our purpose, we need onlythe behaviour at the cusp infinity, which we now recall.


Let π be the period of the Carlitz module (well-defined up to multiplication by anelement in F×q ). The Carlitz exponential e is the holomorphic function C∞ → C∞defined by

e(z) = z∏

λ∈πA−0

(1− z

λ

).

It is surjective and Fq-linear with kernel πA. For z ∈ C∞ −A, let

t(z) = 1e(πz)

= 1π

∑λ∈A

1z − λ

.

The function t, invariant by translations z 7→ z + a (a ∈ A), is then a uniformizer at thecusp infinity. Since any f satisfying (5) is invariant under such translations, it has aLaurent series expansion f(z) =

∑i≥i0 ai(f)t(z)i with i0 ∈ Z (the series does not converge

on all Ω, but only for |t(z)| small enough). Such a function is said to be holomorphic atthe cusp infinity if the expansion has the form

∑i≥0 ai(f)ti. We call it the t-expansion

of f (at infinity). Since Ω is a connected rigid analytic space, any Drinfeld modular formis uniquely determined by its t-expansion.

Let Mk,m(Γ0(n)) be the space of Drinfeld modular forms of weight k and type m forΓ0(n). It is a finite-dimensional vector space over C∞ whose dimension may be calculatedexplicitly thanks to Gekeler [5]. If a0(f) = 0 (resp. a0(f) = a1(f) = 0) and similarconditions at other cusps, f is cuspidal (resp. doubly cuspidal) and the subspace of suchfunctions is denoted by M1

k,m(Γ0(n)) (resp. M2k,m(Γ0(n))). Goss observed that doubly

cuspidal forms play a role similar to classical cusp forms. For an interpretation of Drinfeldmodular forms as differentials on a Drinfeld modular curve, one may refer to Section V.5in [5].

Type and weight are not independent: namely, if k − 2m 6≡ 0 mod (q − 1), the spaceMk,m(Γ0(n)) is zero. Therefore, from now on we assume k ≡ 2m mod (q − 1).

Since Γ0(n) contains matrices of the form(λ 00 1)for λ ∈ F×q , (5) implies ai(f) = 0 when

i 6≡ m mod (q − 1). Thus any f ∈Mk,m(Γ0(n)) has t-expansion of the form∑j≥0

am+j(q−1)(f)tm+j(q−1).

For j ≥ 0, letbj(f) = am+j(q−1)(f).

Later on, we will use both notations for coefficients. A Drinfeld modular form of type> 0 (resp. > 1) is automatically cuspidal (resp. doubly cuspidal). If f is doubly cuspidal,the coefficient b0(f) may not vanish in general (it does when m ∈ 0, 1).

4.2. Hecke algebra. — We define a formal Hecke algebra Rn which acts on thedifferent spaces Mk,m(Γ0(n)). In this section, we adopt the notation Γ = Γ0(n).

Let ∆ = ∆0(n) be the set of matrices(a bc d

)with entries in A such that ad − bc is

monic, c ∈ n and (a) + n = A. Let Rn be the C∞-vector space spanned by formal sumsof double cosets ΓgΓ for g ∈ ∆. One can make Rn a commutative algebra over C∞ (seeSection 3.1 of [17] for a general treatment or Section 6.1 of [3] for Drinfeld modularforms).

10 C. ARMANA

For an ideal p of A, let ∆p be the set of g ∈ ∆ such that det g is the monic generatorof p. The Hecke operator Tp is then defined as the formal sum of all double cosets ΓgΓwith g ∈ ∆p in Rn. For example, when p is prime, Tp = Γ

( 1 00 P)Γ where P is the monic

generator of p.As elements of Rn have coefficients in a field of characteristic p, the usual relation for

the product givesTpTp′ = Tpp′ for any ideals p, p′

(see [11]). This is very different from Hecke operators for classical modular forms, wherethe above relation only holds for relatively prime ideals. One can check that Rn is thepolynomial ring over C∞ spanned by Tp for p prime (such elements are algebraicallyindependent over C∞).

As for the notation, Tp depends on the subgroup Γ0(n) but from the context, therewill be no confusion on which Hecke algebra (or space of endomorphisms of Drinfeldmodular forms) it belongs to.

For n = A, let us consider the formal Hecke algebra RA attached to the sets GL2(A)and ∆0(A). Let Tp temporarily denotes the pth Hecke operator in RA. The map Tp 7→ Tp,for p prime, defines an algebra homomorphism RA → Rn. We regard RA as a universalformal Hecke algebra, independent of n. Any algebraic relation among the Hecke operatorsin RA can be translated to the corresponding relation in Rn for any n.

4.3. Hecke operators on Drinfeld modular forms. — For v =(a bc d

)with entries

in A and f : Ω→ C∞, let

f|[v]k : z 7−→ (ad− bc)k−1(cz + d)−kf(az + b

cz + d

).

Fix g ∈ ∆. The group Γ acts on the left on the double coset ΓgΓ. Let gii be a finitesystem of representatives such that gi has monic determinant. We define an action ofΓgΓ on f ∈Mk,m(Γ0(n)) by

f|[ΓgΓ]k =∑i

f|[gi]k

(independently of the choice of gii). It extends, in a unique way, to a non-faithfulaction of Rn on Mk,m(Γ0(n)). Let T = Tk,m(Γ0(n)) be the commutative sub-C∞-algebraof EndC∞(Mk,m(Γ0(n))) induced by Rn.

For any g ∈ ∆p, a set of representatives of Γ\ΓgΓ with monic determinant is given by(α β0 δ

), α, δ monic in A, (αδ) = p, (α) +A = n, β ∈ A/(δ).

Therefore, the action of Tp on the Drinfeld modular form f can be written more concretelyas

(6) Tp(f)(z) = P k−1 ∑α,δ monic ∈A

β∈A, deg β<deg δαδ=P,(α)+n=A

δ−kf

(αz + β

δ

)= 1P

∑α,β,δ

αkf

(αz + β

δ

)

where P is the monic generator of p. This formula slightly differs from other references.Gekeler [7] (resp. Böckle [3], Section 6) considered PTp (resp. Pm+1−kTp). In particular,our operator coincides with Böckle’s when k = m − 1 (for instance, when k = 2 and


m = 1). In general, these variously normalized Hecke operators have the same eigenforms,however with different eigenvalues.

The Hecke algebra T stabilizes the subspaces M1k,m(Γ0(n)) et M2

k,m(Γ0(n)) (see forexample Proposition 6.9 of [3]). We denote by T′ = T′k,m(Γ0(n)) the restriction of T toM2k,m(Γ0(n)).

5. Hecke action on the first coefficient of Drinfeld modular forms

We recall some results on Goss polynomials for finite lattices and their role in thet-expansion of Drinfeld modular forms. Then we give an explicit formula for the actionof Hecke operators on the first coefficient of this expansion.

5.1. Goss polynomials. — Let Λ be a Fq-lattice in C∞, i.e. a Fq-submodule of C∞having finite intersection with each ball of C∞ of finite radius. We assume Λ to be finiteof dimension d over Fq. The exponential corresponding to Λ

eΛ(z) = z∏

λ∈Λ−0

(1− z

λ

)(z ∈ C∞)

is an entire Λ-periodic Fq-linear function. Since Λ is finite, it is a polynomial in z of theform

eΛ(z) =d∑i=0

λizqi

with coefficients λi ∈ C∞ depending on Λ. Goss has considered Newton’s sums associatedto the reciprocal polynomial of eΛ(X − z) = eΛ(X)− eΛ(z) ∈ C∞[z][X], namely

N0 = 0Nj(z) = Nj,Λ(z) =

∑λ∈Λ

1(z+λ)j (j ≥ 1, z ∈ C∞ − Λ).

LettΛ(z) = eΛ(z)−1 =

∑λ∈Λ

1z − λ

(z ∈ C∞ − Λ).

Proposition 5.1 (Section 2 of [13], 3.4–3.9 in [7]). — Let j ≥ 1. There exists aunique polynomial Gj = Gj,Λ(X) ∈ C∞[X] such that the following equalities hold:

1. if j ≤ q then Gj(X) = Xj

2. Gj(X) = X∑i≥0,j−qi≥0 λiGj−qi(X).

The polynomial Gj(X) is monic of degree j and satisfies Nj = Gj(tΛ). Moreover,

(7) Gj(X) =j−1∑n=0

∑i

(ni

)λiXn+1

for i = (i0, . . . , id) running through (d+ 1)-tuples such that

i0 + . . .+ id = n

i0 + i1q + . . .+ idqd = j − 1

12 C. ARMANA

and λi denotes λi00 · · ·λidd . The polynomial Gj(X) is divisible by Xu where u = bj/qdc+1.

We further put G0,Λ(X) = 0.

Gekeler provided the explicit formula (7) using a generating function.

5.2. Hecke algebra and Goss polynomials. — Let p an ideal of A of degree d ≥ 0with monic generator P . Recall that C denotes the Carlitz module over C∞ (Section3.1). As usual, for an indeterminate X, put CP (X) =

∑di=0CP,iX

qi . For our purpose,we consider the Fq-lattice of dimension d

ΛP = Ker(CP ) = x ∈ C∞ | CP (x) = 0

whose jth Goss polynomial is denoted by Gj,P . Let

tP (z) = t(Pz) = 1e(πPz)

(z ∈ C∞ −A).

Then, if fP (X) is the P th inverse cyclotomic polynomial CP (X−1)Xqd , one has

(8) tP = tqd

fP (t).

The following statement mildly extends Gekeler’s formula 7.3 in [7] (which wasestablished for GL2(A) and p prime) to Γ0(n) and any p.

Proposition 5.2. — Let f ∈Mk,m(Γ0(n)) with t-expansion∑i≥0 ait

i. We have

(9) Tpf = P k−1∑i≥0

∑δ monic in Aδ|P, (P

δ)+n=A

δ−kaiGi,δ(δtPδ)

Moreover, for fixed j, only a finite number of terms in the right-hand side contribute totj in the t-expansion of Tpf .

Proof. — Let δ be a monic divisor of P . Recall that e is the Carlitz exponential. Wewrite F (z) for

∑β∈A,deg β<deg δ f ((Pz/δ + β)/δ). For |t(z)| small enough, F (z) is

∑β∈A,deg β<deg δ

∑i≥0

ait

(Pδ z + β

δ

)i=∑i≥0

ai∑

β∈A,deg β<deg δe

(πPδ z + β

δ

)−i

=∑i≥0

ai∑

β∈A,deg β<deg δ

(e

(πPz

δ2

)+ e

(πβ

δ

))−iby additivity of e. According to the analytic theory of Drinfeld modules, the finite sete(πβ/δ) | β ∈ A,deg β < deg δ is in bĳection with the lattice Λδ = Ker(Cδ). Letw = Pz/δ2. Then, by Proposition 5.1, F (z) is∑

i≥0ai∑λ∈Λδ

(e(πw) + λ)−i =∑i≥0

aiNi,Λδ(e(πw)) =∑i≥0

aiGi,Λδ(eΛδ(e(πw))−1).

Observe that eΛδ(z) = Cδ(z)/δ (both polynomials have the same set of zeros and themultiplicative constant is obtained by comparing the terms in z). By the properties of


the Carlitz exponential, we also have Cδ(e(πw)) = C(πzP/δ) = t(zP/δ)−1. Substituting,we get

F (z) =∑i≥0

aiGi,Λδ

(δt

(zP

δ

))=∑i≥0

aiGi,Λδ(δtPδ(z)).

Our claim follows from (6) and the last statement of Proposition 5.1.

Remark 5.3. — To obtain the t-expansion of Tpf from Equation (9), it would sufficeto compose the t-expansions of tP/δ and Goss polynomials Gi,δ. However, making thisexplicit seems to be a difficult problem. Indeed, a similar question arises when trying tomake explicit the t-expansion of Drinfeld-Eisenstein series (see Section 6 of [7]) since itinvolves the t-expansion of Gi,πA(tP )(1). This is quite different from the classical situationwhere coefficients of Eisenstein series are well-known arithmetic functions.

5.3. Hecke module spanned by b1. —

Notation 5.4. — The dual space of Mk,m(Γ0(n)) has the natural right action of T,given by composition, and contains the following linear forms, for any n ≥ 1:

am+n(q−1) = bn : f 7→ am+n(q−1)(f) = bn(f).

Let u = uk,m,n : T′ → HomC∞(M2k,m(Γ0(n)),C∞) be the C∞-linear map s 7→ b1s. We

write b1T′ for the image of u.

We collect some remarks on the dimension of the C∞-algebra T′. The map u is notnecessarily an isomorphism, therefore the dimension of T′ is unknown a priori. In the caseT′ = T′2,1(Γ0(n)), one can prove that its dimension coincides with dimC∞M

22,1(Γ0(n)),

using results from automorphic forms and work of Gekeler and Reversat [15].We keep Notation 3.3. The next statement gives a first description of b1T′.

Proposition 5.5. — Let f ∈Mk,m(Γ0(n)) with t-expansion∑i≥0 ai(f)ti. Let p an ideal

of A of degree d with monic generator P . Then am+(q−1)(Tpf) is

(10)∑n

(m+q−2n

)C(P )na1+n0+n1q+...+ndqd(f) + ε

∑Q|P,Q∈A1+(Q)+n=A

Qk−1a1(f)

where n = (n0, . . . , nd) is such that n0 + . . .+ nd = m+ q − 2 with ni ≥ 0 for all i and ε

is defined by ε =

1 if m = 10 otherwise.

Remark 5.6. — 1. In Example 7.4 of [7], Gekeler treated ai(Tpf) for p of degree 1,i ≥ 0, and f modular for GL2(A). Proposition 5.5 supplements Gekeler’s statement.

2. Actually, Propositions 5.2 and 5.5 work for holomorphic functions f : Ω → C∞having an expansion

∑i≥0 ait

i for |t(z)| small enough (Hecke operators are stilldefined on f via (6)). In particular, this applies to holomorphic functions f : Ω→C∞ which are A-periodic (f(z + a) = f(z), a ∈ A) and holomorphic at the cuspinfinity.

(1)The lattice πA is not finite but Goss polynomials can be defined in that more general setting (see[13, 7]).

14 C. ARMANA

Proof. — By Proposition 5.2, we have to find the coefficient of tm+(q−1) in the t-expansionof Gi,δ(δtP/δ). First, if i = 0, then G0,δ(X) = 0 so the expansion of G0,δ(δtP/δ) has notm+(q−1)-term.

Assume i > 0. By (8) the t-expansion of tP/δ is divisible by tqd−deg δ . Moreover, itfollows from the definition of Goss polynomials that Gi,δ(X) has X as a factor. Hence,the t-expansion of Gi,δ(δtP/δ) is divisible by tqd−deg δ . Since m < q− 1, Gi,δ(δtP/δ) has notm+(q−1)-term when d− deg δ ≥ 2. Now assume 0 ≤ d− deg δ ≤ 1. Put s = deg δ. Recallthat eΛδ(z) = Cδ(z)/δ =

∑si=0Cδ,iz

qi/δ. The explicit formula for Goss polynomials gives

Gi,δ(X) =i−1∑j=0

δ−j∑n

(jn

)C(δ)nXj+1

where n = (n0, . . . , ns) are such that n0 + . . .+ ns = j and n0 + n1q + . . .+ nsqs = i− 1.

Suppose that s = d, i.e. δ = P . Then the corresponding partial sum in (9) is

1P

∑i≥0

aiGi,P (Pt) = 1P

∑i≥0

ai

i−1∑j=0

P−j∑n

(jn

)C(P )n(Pt)j+1.

The tm+(q−1)-term corresponds to j = m+ q − 2; namely, it is∑n

(m+q−2n

)C(P )na1+n0+n1q+...+ndqd(f)

with n = (n0, . . . , nd) such that n0 + . . .+ nd = m+ q − 2.Next, suppose that s = d− 1. Using Equation (8), we get

(11) Gi,δ(δtPδ) =

i−1∑j=0

δ−j∑n

(jn

)C(δ)n

(δ

tq

1 + Pδ tq−1

)j+1

where (n0, . . . , nd−1) with n0 + . . .+ nd−1 = j and n0 + n1q + . . .+ nd−1qd−1 = i− 1. If

j ≥ 1, then q(j + 1) ≥ 2q > m+ q− 1, thus there is no tm+(q−1)-term in the expansion of(11). Finally, we assume j = 0, in other words n0 = . . . = nd−1 = 0 and i = 1. We have

G1,δ(δtPδ) = δ

tq

1 + Pδ tq−1 = δtq

+∞∑n=0

(−1)nPn

δntn(q−1).

This series has a tm+(q−1)-term if and only if q − 1 divides m− 1. This happens only ifm = 1, and in that case the corresponding coefficient is δ. Thus we obtain (10).

Assume m ∈ 0, 1. By (10), the linear form b1Tp = am+(q−1)Tp is a A-linear combina-tion of ai, where i runs through the set of natural integers satisfying the condition: theexpansion of i in base q has at most d+ 1 digits, whose sum is equal to m+ q − 1. Inparticular, the set of such i’s only depends on the degree d of p. This observation, alsocommunicated to the author by D. Goss, will be used in Section 7. For the moment, wederive the following statement for b1T′.

Notation 5.7. — Let S be the set of natural integers of the form c/(q − 1) wherec ∈ N is such that the sum of its base q digits is q − 1.


Corollary 5.8. — If m ∈ 0, 1 then b1T′ is contained in the C∞-vector space spannedby bn for n ∈ S .

The reverse inclusion will be proved in Section 7. Finally, we state another straightfor-ward application of Proposition 5.5.

Notation 5.9. — For d ≥ 1 and i = (i0, . . . , id−1), let

Θd(i0, . . . , id−1) =∑

P∈Ad+

C(P )iTP =∑

P∈Ad+

Ci0P,0 · · ·Cid−1P,d−1TP ∈ RA.

Corollary 5.10. — Let f ∈ Mk,m(Γ0(n)). With the notations of Proposition 5.5 andSection 3, the coefficient am+(q−1)(Θd(i0, . . . , id−1)f) is∑

n=(n0,...,nd)n0+...+nd=m+q−2

(m+q−2n

)Sd(n0 + i0, . . . , nd−1 + id−1)a1+n0+n1q+...+ndqd(f)

+ε∑

P∈Ad+

C(P )i∑

Q|P,Q∈A1+(Q)+n=A

Qk−1a1(f)

where ε is defined as in Proposition 5.5.

6. Annihilator of b1 for the Hecke action

Notation 6.1. — Let I = Ik,m,n be the kernel of u i.e. the ideal of elements s ∈ T′such that b1s = 0 in the dual space of M2

k,m(Γ0(n)).

In particular, I is a sub-C∞-algebra of T′ which maps doubly cuspidal forms toDrinfeld modular forms f satisfying a0(f) = b0(f) = b1(f) = 0.

Lemma 6.2. — If the map u : T′ → HomC∞(M2k,m(Γ0(n)),C∞) is surjective, then it

is an isomorphism.

Proof. — Since u is surjective, we take an element tn in the preimage of bn for anyn ≥ 1. If s belongs to the ideal I, so does tns. Hence, for any f ∈ M2

k,m(Γ0(n)), thenth coefficient bn(sf) is zero for any n ≥ 1. As the t-expansion uniquely determines aDrinfeld modular form, sf must be zero. Therefore s is zero as an endomorphism ofM2k,m(Γ0(n)).

6.1. Proof of Theorem 1.1. —

Proof of Theorem 1.1. — Actually we prove a slightly more general statement: all thefollowing equalities of linear forms will take place in the dual space of Mk,m(Γ0(n)) ifm 6= 1 (resp. of M2

k,m(Γ0(n)) if m = 1).1. Without any assumption on m, we apply Corollary 5.10 to d = 1. For i ≥ 0 we get

b1

∑P∈A1+

P iTP

=m+q−2∑n=0

(m+q−2n

)S1(n+ i)a1+n+q(m+q−2−n).

16 C. ARMANA

This follows also from Gekeler’s example 7.4 in [7], although stated there for GL2(A)and with a different normalization of Hecke operators.

Assume m = 0. The sum S1(n + 1) =∑Q∈A1+ Q

n+1 is nonzero if and only ifn = q − 2, and S1(q − 1) = −1 (by Lemma 3.4 for instance). Taking i = 1, ourexpression simplifies as b1

(∑P∈A1+ PTP

)= −b1.

Assume m = 1. Since the sum S1(n) is nonzero if and only if n = q − 1, takingi = 0, we get b1

(∑P∈A1+ TP

)= −b1.

2. Consider (i0, . . . , id−1) as in the statement. By Corollary 5.10, we get thatb1(Θd(i0, . . . , id−1)) is∑

nn0+...+nd=m+q−2

(m+q−2n

)Sd(n0 + i0, . . . , nd−1 + id−1)a1+n0+n1q+...+ndqd .

We have 0 ≤ nj + ij ≤ 2(q − 1), hence we can evaluate Sd(n0 + i0, . . . , nd−1 + id−1)thanks to Proposition 3.5. This sum is nonzero if and only if nj + ij = q − 1 or2(q − 1) for all j ∈ 0, . . . , d− 1. If this happens, we have

d(q − 1) ≤d−1∑l=0

(nl + il) ≤ i0 + . . .+ id−1 +m+ q − 2

which contradicts i0 + . . .+ id−1 ≤ (d− 1)(q− 1)−m. Accordingly, the sum alwaysvanishes and b1(Θd(i0, . . . , id−1)) = 0.

3. Apply the statement proved before to i0 = l and i1 = . . . = id−1 = 0.

It is worth pointing out that the elements of I given in Theorem 1.1 are universal inthe sense that, for a given type, they do not depend on the weight k nor the ideal n. Someof them, as

∑P∈Ad+

TP for d ≥ 2 for instance, are also independent of the type m. Thismeans that, in the universal formal Hecke algebra RA, such an element is independent ofk, m and n.

Remark 6.3. — This phenomenon does not occur for classical modular forms of weight2 as we now explain. Let S2(Γ0(N)) be the complex space of weight-2 cusp forms forΓ0(N) (N ≥ 1). We write (cn)n≥1 for the linear forms given by Fourier coefficients ofsuch modular forms at the cusp infinity. The Hecke algebra Tc of weight 2 for Γ0(N)is the subring of End(S2(Γ0(N)) spanned over C by all Hecke operators Tn for n ∈ N.Let uc be the C-linear map Tc → HomC(S2(Γ0(N),C) given by s 7→ c1s. Relation (3)gives cn = uc(Tn) for all n ≥ 1, thus uc is bĳective. We claim that if there exists aC-linear combination s = λ1Ti1 + . . .+ λjTij , with j, λ1, . . . , λj , i1, . . . , ij independent ofN , such that s = 0 as an endomorphism of S2(Γ0(N)), then the coefficients λ1, . . . , λsmust be zero. In fact, when N is prime, the Hecke operators T1, . . . , Tg(N) are C-linearlyindependent in End(S2(Γ0(N)) for g(N) = dimS2(Γ0(N)) (this follows from the cuspinfinity not being a Weierstrass point on the modular curve X0(N)). Choosing N primesuch that g(N) is large enough yields λ1 = . . . = λj = 0 and proves our claim.

In Section 7.2, we will further our investigation of the ideal I and prove that it vanishesin some cases (Theorem 7.7).


6.2. Linear relations for eigenvalues. —

Notation 6.4. — Let p an ideal of A with monic generator P . A Hecke eigenform f isa Drinfeld modular form which is an eigenform for all Hecke operators. We write λP (f)for its eigenvalue for TP = Tp.

For a Hecke eigenform f such that b1(f) 6= 0, Theorem 1.1 yields linear relationsamong its eigenvalues. It seems rather remarkable that these relations are universal inthe sense that, for a fixed type, they do not depend on the weight k nor on the level n.

Proposition 6.5. — Let f ∈ Mk,m(Γ0(n)) be a Hecke eigenform with b1(f) 6= 0. Ifm = 1, we assume further f ∈M2

k,m(Γ0(n)).1. If m ∈ 0, 1, then ∑

P∈A1+

P 1−mλP (f) + 1 = 0.

2. Let d ≥ 1 and i0, . . . , id−1 satisfying (1) and (2). Then∑P∈Ad+

C(P )iλP (f) = 0.

3. Let l and d be integers such that 0 ≤ l ≤ q −m and d ≥ (l +m)/(q − 1) + 1. Then∑P∈Ad+

P lλP (f) = 0.

In particular, if d ≥ 2, or f has type 0 and d = 1, then∑P∈Ad+

λP (f) = 0.

6.3. Linear relations for Hecke operators. — We explain how some relations ofProposition 6.5 may follow from linear relations among Hecke operators in characteristiczero or p. In other words, we prove or suggest that certain elements of I given in Theorem1.1 are zero in T′.

Notation 6.6. — For an ideal n of A, let Hn be the abelian group of Z-valued cuspidalharmonic cochains for Γ0(n) on the Bruhat-Tits tree T of PGL(2,K∞) (we refer toSection 3 of [15] for the relevant definitions and properties). The group GL2(K) acts onthe left on the set of oriented edges Y (T ) of T . We define an endomorphism θp of Hn by

(θpF )(e) =∑

α,δ monic ∈Aβ∈A, deg β<deg δ(αδ)=p, (α)+n=A

F

((α β0 δ

)e

)

for F ∈ Hn and e ∈ Y (T ).

After scalar extension to the complex numbers C, Hn is identified with a space ofcuspidal automorphic forms on GL(2) over the adeles of K (by the strong approximationtheorem). Moreover, using Teitelbaum’s residue map [19], Gekeler and Reversat [15]gave an isomorphism between Hn/pHn and a subspace of Drinfeld modular forms, namelythe subspace M2

2,1(Γ0(n),Fp) of M22,1(Γ0(n)) consisting of such forms with residues in Fp.

18 C. ARMANA

It turns out that this isomorphism is Hecke-equivariant, with the normalizations we haveadopted here for Tp and θp. Finally, M2

2,1(Γ0(n),Fp) is an Fp-vector space which, afterscalar extension to C∞, gives the whole space M2

2,1(Γ0(n)). Put differently, the Heckeoperator Tp acting on M2

2,1(Γ0(n)) can be thought of as the mod p reduction of θp.

Lemma 6.7. — Let n be a prime. Assume d ≥ deg(n) − 1. Then∑

deg p=d θp = 0. Inparticular,

∑deg p=d Tp = 0 on M2

2,1(Γ0(n)).

Proof. — Let F ∈ Hn(C) = Hn ⊗Z C be an eigenform for (θp)p with eigenvalues (λp)p.For d > deg(n)− 3, we have

(12)∑

deg p≤dλp = 0.

It is essentially a consequence of the cuspidality of F . Namely, by the structure of thequotient graph Γ0(n)\T , the edges of T corresponding to matrices

(πk 00 1

)with k ≥ deg n

are not in the support of F (see also [18]). Using the Fourier expansion of F and therelation between Fourier coefficients and Hecke eigenvalues (λp)p ((3.12’) and (3.13)in [8]), we derive (12). Since n is prime, there exists a basis of Hn(C) consisting ofnormalized eigenforms for (θp)p. Hence we have

∑deg p≤d θp = 0 if d > deg(n)− 3. An

equivalent formulation is:∑

deg p≤deg(n)−2 θp = 0 and∑

deg p=d θp = 0 if d ≥ deg(n) − 1.This completes the proof.

Therefore, from the theory of automorphic forms, we know that certain elements of Igiven in Theorem 1.1 are zero on M2

2,1(Γ0(n)), because so they are on Hn: this is thecase for

∑deg p=d Tp if n is prime and d ≥ deg(n)− 1.

It is now natural to ask whether some elements of I in Theorem 1.1 can act nontriviallyon Hn and be zero in T′ (i.e. in characteristic p). We suggest that this happens.

Question 6.8. — Assume n is prime. Do the following relations among Hecke operatorson M2

2,1(Γ0(n)):1.∑

deg p≤1 Tp = 0 if n has degree 42.∑

deg p=deg(n)−2 Tp = 0 if n has degree ≥ 4hold?

We checked numerically such relations on several examples. We computed Heckeoperators on Hn/pHn, for n prime, using Teitelbaum’s modular symbols for Fq(T ) [20].The first relation has been checked for q ∈ 2, 3, 4, 5, 7 and the second one for all primesn of degree 5 and 6 in F2[T ]. Note that, when deg n = 4, both relations are equivalent:indeed, we have

∑deg p≤2 θp = 0 (see proof of Lemma 6.7).

An affirmative answer to Question 6.8 would tell that some elements of I would bezero in T′ but may be nonzero on the automorphic level, more precisely:•∑

deg p≤1 Tp = 0 in T′2,1(Γ0(n)) for n prime of degree 4;•∑

deg p=deg(n)−2 Tp = 0 in T′2,1(Γ0(n)) for n prime of degree ≥ 4.In the next paragraph, we are interested in the reverse problem: finding nonzero

elements in the ideal I.


6.4. Nonzero elements in the annihilator. — The following conjecture suggeststhat, in general, the Hecke annihilator I of b1 is nonzero.

Conjecture 6.9. — Assume n is prime of degree ≥ 5. Then∑

deg p≤1 Tp ∈ I is nonzeroas an endomorphism of M2

2,1(Γ0(n)). In particular, the map

u : T′ −→ HomC∞(M22,1(Γ0(n)),C∞)

s 7−→ b1s

is not surjective.

The last statement follows from Lemma 6.2. As in Section 6.3, we were able to computethe action of

∑deg p≤1 Tp on M2

2,1(Γ0(n)) on some examples. We checked Conjecture 6.9for all primes n in F2[T ] of degree in 5, 6, 7, 8, 9, in F3[T ] of degree in 5, 6, 7, 8, inF4[T ] and F5[T ] of degree 5 .

7. Proof and applications of Theorem 1.2

7.1. Explicit version of Theorem 1.2. —

Notation 7.1. — We call a decomposition of c ∈ N a tuple c = (c0, . . . , cd) such thatc =

∑dj=0 cjq

j and 0 ≤ cj < q for any j ∈ 0, . . . , d, for some d ≥ 0. The length of c isd+ 1. Note that we do not require cd 6= 0. The base q expansion gives a decompositionof c. By putting zeros at the end of any decomposition of c, we obtain decompositions oflarger length.

If i = (i0, . . . , id) is a decomposition of i ≥ 0, let

l(i) =∑

P∈Ad+

C(P )i∑

Q|P,Q∈A1+(Q)+n=A

Qk−1 ∈ A.

We prove Theorem 1.2 by establishing the following explicit version.

Theorem 7.2. — Assume q is a prime.1. Suppose m = 0. Let n = c/(q − 1) ∈ S . We fix a decomposition (c0, . . . , cd) of c of

length d+ 1 for some d ≥ 0 (therefore c0 + . . .+ cd = q − 1). Let

tc0,...,cd = (−1)d( q−2c0−1,c1,...,cd

)−1 ∑P∈Ad+

Pqd+1−c

TP ∈ RA.

Then, for any k and n, we have bn = b1tc0,...,cd in the dual space of Mk,0(Γ0(n)).2. Suppose m = 1. Let n = c/(q − 1) ∈ S . We fix a decomposition (c0, . . . , cd) of c of

length d+ 1 for some d ≥ 0 with cd 6= q − 1 (therefore c0 + . . .+ cd = q − 1). Let

t′c0,...,cd = (−1)d( q−1c0,...,cd

)−1 ∑P∈Ad+

Pqd+1−1−c

TP ∈ RA.

Then, for any k and n, we have

bn = b1t′c0,...,cd + (−1)d+1( q−1

c0,...,cd

)−1l(qd+1 − 1− c)a1

in the dual space of Mk,1(Γ0(n)).

20 C. ARMANA

3. Assume m = 1. Let d ≥ 1 and

td = (−1)d∑

P∈Ad+

( Pqd−1

−d−1∑i=0

Pqd−1−(q−1)qi

)TP ∈ RA.

Then, for any k and n, we have

(13) bqd = b1td + (−1)d(−l(qd − 1) +

d−1∑i=0

l(qd − 1− (q − 1)qi))a1

in the dual space of Mk,1(Γ0(n)).

Remark 7.3. — 1. Since q is prime and∑dj=0 cj = q−1, the multinomial coefficients( q−1

c0,...,cd

)and

( q−2c0−1,c1,...,cd

)are nonzero in Fp, by Lucas’s theorem, hence invertible.

2. On doubly cuspidal forms, a1 vanishes and the expressions of Theorem 7.2 simplifyand provide Theorem 1.2. Moreover, since b1T′ is contained in the C∞-vector spacespanned by bn for n ∈ S (Corollary 5.8), we get the equality provided that q isprime and m ∈ 0, 1.

3. For a given n ∈ S , we get infinitely many expressions sn ∈ T′ such that bn = b1sn.The reason is that, in the first two items of Theorem 7.2, any decomposition ofc = (q−1)n gives rise to a formula for sn ∈ T′ satisfying the desired property. Moregenerally, any element of sn + I would satisfy the same property.

4. The primality assumption on q is not always essential: it is required to ensure thatthe multinomial coefficient

( q−1c0,...,cd

)for m = 1 (resp.

( q−2c0−1,c1,...,cd

)for m = 0) is

nonzero in Fp. Hence, the assumption is unnecessary in (13). If q is not a prime,the first (resp. second) statement of Theorem 7.2 is true for n = c/(q − 1) ∈ Ssuch that there exists a decomposition (c0, . . . , cd) of c with

( q−2c0−1,c1,...,cd

)6= 0 in Fp

(resp.( q−1c0,...,cd

)6= 0 in Fp) for some d ≥ 0.

Before proving Theorem 7.2, we give an example.

Example 7.4 (d = 1). — We put

sn = −(q−1n−1

)−1 ∑P∈A1+

Pn−1TP for 1 ≤ n ≤ q − 1

sq = −∑

P∈A1+

(P q−1 − 1)TP .

Then bn(f) = b1(sn(f)) for all f ∈ M2k,1(Γ0(n)) and 1 ≤ n ≤ q. This is valid for q a

power of a prime, by Remark 7.3 and Lucas’s theorem. Using these formulas, we canrecover the first q coefficients of any Hecke eigenform f in M2

k,1(Γ0(n)) in terms of b1(f)and the eigenvalues.

Proof of Theorem 7.2. — 1. Assume that the type m is 0. We put n0 = c0 − 1, n1 =c1, . . . , nd = cd, so that n0 + . . .+ nd = q − 2. By Corollary 5.10, aq−1(Θd(q − 1−n0, . . . , q − 1− nd−1)) is∑

r

(q−2r

)Sd(r0 + q − 1− n0, . . . , rd−1 + q − 1− nd−1)a1+r0+r1q+...+rdqd


where r = (r0, . . . , rd) satisfies r0 + . . . + rd = q − 2. From ni ≥ −1, we get0 ≤ ri + q − 1 − ni ≤ 2(q − 1) for all i. We can thus evaluate the sum Sd(r0 +q − 1 − n0, . . . , rd−1 + q − 1 − nd−1) by Proposition 3.5: it is nonzero only if r issuch that ri ∈ ni, q − 1 + ni, for all i ∈ 0, . . . d− 1. Since ri ≤ q − 2, we haver0 = n0, . . . , rd−1 = nd−1 and by Proposition 3.5,

aq−1(Θd(q − 1− n0, . . . , q − 1− nd−1)) =(q−2n

)(−1)da1+n0+n1q+...+ndqd .

Finally, a1+n0+...+ndqd = an(q−1) = bn and the conclusion follows.2. Assume that the typem is 1. Since qd+1−1−c has base q expansion

∑dj=0(q−1−cj)qj ,

we have ∑P∈Ad+

Pqd+1−1−c

TP =

∑P∈Ad+

Cq−1−c0P,0 . . . C

q−1−cd−1P,d−1 TP

= Θd(q − 1− c0, . . . , q − 1− cd−1).

By Corollary 5.10, b1(Θd(q − 1− c0, . . . , q − 1− cd−1)) is∑r

(q−1r

)Sd(r0 + q − 1− c0, . . . , rd−1 + q − 1− cd−1)a1+r0+r1q+...+rdqd

+ l(qd+1 − 1− c)a1

with r = (r0, . . . , rd) such that r0+. . .+rd = q−1. From ci ≥ 0 and 0 ≤ ri ≤ q−1, weget 0 ≤ ri+q−1−ci ≤ 2(q−1). Thus the sum Sd(r0+q−1−c0, . . . , rd−1+q−1−cd−1)can be evaluated thanks to Proposition 3.5: it is nonzero if and only if ri ∈ci, q − 1 + ci for all i ∈ 0, . . . , d− 1.

Suppose there exists k ∈ 0, . . . , d− 1 with rk = q − 1 + ck. Then, according tothe previous remarks, we have

q − 1− rd =d−1∑j=0

rj = q − 1 + ck +d−1∑

j=0,j 6=krj ≥ q − 1 +

d−1∑j=0

cj = 2(q − 1)− cd

hence 0 ≤ q−1−cd ≤ −rd. This implies rd = 0, thus cd = q−1, which is impossible.Therefore, we have rj = cj for any j ∈ 0, . . . , d− 1 and rd = cd as a consequence.Proposition 3.5 then provides

b1(Θd(q − 1− c0, . . . , q − 1− cd−1)) =a1+c0+c1q+...+cdqd

+ (−1)d( q−1c0,...,cd

)−1l(qd+1 − 1− c)a1.

Finally, a1+c0+c1q+...+cdqd = a1+n(q−1) = bn, thus the statement is proved.3. Assume that the type m is 1. We first compute b1(Θd(q − 1, . . . , q − 1)). According

to Corollary 5.10, it is∑r=(r0,...,rd)

r0+...+rd=q−1

(q−1r

)Sd(r0 + q − 1, . . . , rd−1 + q − 1)a1+r0+r1q+...+rdqd

+ l(qd+1 − 1)a1.

By Proposition 3.5, the sum Sd(r0 + q − 1, . . . , rd−1 + q − 1) is nonzero if and onlyif ri ∈ 0, q − 1 for all i ∈ 0, . . . , d− 1. This means that (r0, . . . , rd−1) is one of

22 C. ARMANA

the following:

(q − 1, 0, . . . , 0) , (0, q − 1, 0, . . . , 0) , . . . , (0, . . . , 0, q − 1) , (0, . . . , 0).

Thus b1(Θd(q − 1, . . . , q − 1)) equals

(14) (−1)d(a1+(q−1) + . . .+ a1+(q−1)qd−1 + a1+(q−1)qd

)+ l(qd+1 − 1)a1.

Next, we compute b1(Θ(q − 1, . . . , 0, . . . , q − 1)), the only zero term being at the(j + 1)th position (0 ≤ j ≤ d− 1). From Corollary 5.10, it is∑

r=(r0,...,rd)r0+...+rd=q−1

(q−1r

)Sd(r0 + q − 1, . . . , rj , . . . , rd−1 + q − 1)a1+r0+r1q+...+rdqd

+ l(qd+1 − 1− (q − 1)qj)a1.

Again by Proposition 3.5, the sum is only over r satisfying the following twoproperties:

ri ∈ 0, q − 1 for all i ∈ 0, . . . , d− 1, i 6= j

rj ∈ q − 1, 2(q − 1).

Since r0 + . . .+ rd = q − 1, we have necessarily rj = q − 1, ri = 0 for all i 6= j andrd = 0. Then

(15) b1(Θ(q − 1, . . . , 0, . . . , q − 1)) = (−1)da1+(q−1)qj + l(qd+1 − 1− (q − 1)qj)a1

Combining (14) and (15), we get the claim.

7.2. Applications. — Theorem 1.2 has the following straightforward consequence.

Corollary 7.5. — Under the assumptions of Theorem 1.2, if f is a Hecke eigenformwith bn(f) 6= 0 for some n ∈ S , then b1(f) 6= 0.

In particular, in Proposition 6.5, one can replace the assumption b1(f) 6= 0 by: thereexists n ∈ S such that bn(f) 6= 0.

We now provide multiplicity one statements in certain spaces of Drinfeld modularforms.

Lemma 7.6. — 1. Let d = dimMk,m(GL2(A)). The C∞-linear map

Mk,m(GL2(A)) −→ Cd∞

f 7−→ (b0(f), . . . , bd−1(f))

is an isomorphism.2. Let d = dimM2

2,1(Γ0(n)). The C∞-linear map

M22,1(Γ0(n)) −→ Cd

∞f 7−→ (b1(f), . . . , bd(f))

is an isomorphism.


Proof. — The first assertion follows readily from a formula relating, for a nonzerof ∈Mk,m(GL2(A)), the orders of vanishing of f at elliptic, non-elliptic points and thecusp infinity of GL2(A) (see (5.14) in Gekeler’s paper [7]).

For the second assertion, we consider the Drinfeld modular curve X0(n) attached toΓ0(n). This smooth projective algebraic curve over C∞ is the compactification of theaffine curve Y0(n) = Γ0(n)\Ω over C∞, the group Γ0(n) acting on Ω via linear fractionaltransformations. Actually, the curve Y0(n) is a coarse moduli scheme for rank-2 Drinfeldmodules with a level structure determined by n. The cusps, i.e. the set X0(n)− Y0(n), isnaturally in bĳection with Γ0(n)\P1(K). Since we assume n prime, the cusps are labeled0,∞ as usual. Gekeler gave formulas for the genus g = g(X0(n)) in terms of the degreeof n ([5, 6]).

One can show that ∞ is not a Weierstrass point on X0(n) with n prime (i.e. everyholomorphic differential form on X0(n) vanishes at ∞ at order < g). This is merely anadaptation of Ogg’s geometric argument [16] for the classical modular curve X0(pM)with p prime and p - M . To adapt the proof, we need Gekeler’s description of the badfiber at n of a model of X0(n) over A ([6] p. 233): it consists of two copies of the projectiveline over Fn = A/n intersecting transversally at n supersingular points (i.e. points whoseunderlying rank-2 Drinfeld module is supersingular over Fn). The full Atkin–Lehnerinvolution wn interchanges the components. The reductions of the cusps are on distinctcomponents hence are not supersingular points. The second ingredient is the analogue ofOgg’s formula (2) in [16]: the number n of supersingular points is 1 + g, according to(5.4) in [6] and Gekeler’s formula for g. We leave the details to the reader.

The map f 7→ f(z)dz defines an isomorphism between M22,1(Γ0(n)) and the space of

holomorphic differential forms on X0(n) ([15] Proposition 2.10.2), hence both spaceshave dimension g = d. Since ∞ is not a Weierstrass point, any Drinfeld modularform in M2

2,1(Γ0(n)) vanishes at ∞ at order < d. In other words, the linear mapM2

2,1(Γ0(n))→ Cd∞ given by f 7→ (b1(f), . . . , bd(f)) is injective, hence bĳective.

Theorem 7.7. — Let M be one of the following spaces of Drinfeld modular forms:1. M1

k,0(GL2(A)) with k < (q + 1)2(q − 1)2. M2

k,1(GL2(A)) with k < q2(q + 1)3. M2

2,1(Γ0(n)) with n prime of degree 3.Then:

– Any eigenform in M for the operators (Tp)deg p=1 is characterized in the space Mby its eigenvalues, up to a multiplicative constant.

– The map u : T′ → HomC∞(M,C∞) is an isomorphism.

Proof. — Consider the first two cases for M . By the cuspidality (resp. doubly cus-pidality) condition and the assumption on the type, we have b0(f) = am(f) = 0.Therefore, any function f ∈ M is determined, in the space M , by its coefficientsb1(f), . . . , bd−1(f), according to Lemma 7.6. Now, if f is an eigenform for (Tp)deg p=1,we know that b1(f), . . . , bq(f) are determined by the eigenvalues (up to a multiplica-tive constant), thanks to Example 7.4. Recall that the dimension of Mk,m(GL2(A)) isd =

⌊(k − (q + 1)m)/(q2 − 1)

⌋+ 1 (this follows from Gekeler’s formula (5.14) in [7]).

Here, the assumptions on the weight k ensure that d− 1 ≤ q. The conclusion follows.

24 C. ARMANA

The proof of the third case is similar, except that the dimension of M is q. Indeed,this dimension is equal to the genus of X0(n). By Gekeler’s formula for the genus ([5, 6]),it is q when n is prime of degree 3.

For the bĳectivity of u, we need only to prove the surjectivity by Lemma 6.2. Considerthe first two cases for M . As before, M has dimension d− 1 ≤ q. Moreover, the imageof u contains b1, . . . , bd−1 (by Theorem 7.2) which are linearly independent (by Lemma7.6), hence the conclusion. The proof of the third case is similar.

As a corollary, we get that the dimension of the C∞-algebra T′ coincides with thedimension of the space of Drinfeld modular forms M , for M as in the statement.

7.3. Comment on A-structures. — Although we worked with C∞-structures, mostof the results of this paper could be transferred over the ring A. For instance, one couldwork with the subspace M2

k,m(Γ0(n);A) ⊂M2k,m(Γ0(n)) consisting of modular forms with

expansion in A[[t]] and the Hecke algebra T′A spanned over A by Hecke operators. UsingProposition 5.2, one may check that the map

T′A → HomA(M2k,m(Γ0(n);A), A)

induced by s 7→ b1s, is well-defined. We expect that M2k,m(Γ0(n);A) is a A-structure of

M2k,m(Γ0(n)) (i.e. there exists a basis of M2

k,m(Γ0(n)) consisting of modular forms withcoefficients in A). However, a general theory of such algebraic Drinfeld modular formsis still missing in the literature. Some instances of such a theory can be found in [12](Section 2, for Mk,m(GL2(A))) and [1] (Section 4.2, for M2

2,1(Γ0(n))).

8. Coefficients of h

We use Theorem 7.2 to compute explicitly some coefficients of Gekeler’s Drinfeldmodular form h, defined in [7]. Recall that h has weight q + 1 and type 1 for GL2(A).It is defined as a certain Poincaré series and is also a (q − 1)th root of the Drinfelddiscriminant form ∆. Moreover, it is a cuspidal Hecke eigenform with Tph = h for anyp (Corollary 7.6 in [7] with a different normalization of Hecke operators). The first

coefficients of h are a1(h) = −1 and b1(h) = aq(h) =

0 if q > 21 if q = 2

.

Proposition 8.1. — For P in A, let σP =∑Q|P,Q∈A1+ Q

q.

1. Assume q is a prime > 2. Let c ∈ N such that c =∑dj=0 cjq

j with 0 ≤ cj < q,∑dj=0 cj = q − 1 and cd 6= q − 1 (we do not necessarily assume cd 6= 0). Then

(16) b cq−1

(h) = (−1)d( q−1c0,...,cd

)−1 ∑P∈Ad+

Pqd+1−1−c

σP .

Moreover, for d ≥ 0,

(17) bqd(h) = (−1)d+1 ∑P∈Ad+

(− Pqd−1

+d−1∑i=0

Pqd−1−(q−1)qi

)σP


2. Assume q = 2. Then for every d ≥ 0, one has

(18) b2d(h) = (−1)d∑

P∈Ad+

(− P

2d−1

+d−1∑i=0

P2d−1−2i

)(1 + σP ).

Remark 8.2. — We recover that the corresponding coefficients of h are polynomials inT q − T with coefficients in Fq (indeed, they are elements of A which are invariant underT 7→ T + c for c ∈ Fq). More generally, Gekeler proved that this property holds for anycoefficient of h (Theorem 2.4 of [9]).

Taking d = 1 in Proposition 8.1, one can recover the first q coefficients of h. If qis a prime > 2, then bi(h) = 0 if 1 ≤ i ≤ q − 2, bq−1(h) = −1 and bq(h) = T q − T .They can also be obtained from the Taylor series h = −tU−1

1 + o(t1+(q−1)(q3−q2)) withU1 = 1− t(q−1)2 + (T q − T )t(q−1)q (see Corollary 10.4 in [7]).

For i ∈ N, let [i] = T qi − T . Using congruences and estimates on the degree of

coefficients of h, Gekeler proved that for any d ≥ 1,

(19) bqd(h) =

[d] if q > 21 + [d] if q = 2

(see Corollary 2.6 of [9]; note that his bi denotes our −bi). Equation (17) thus providesan alternative formula for bqd(h). We have not been able to recover (19) from (17) and(18). Hence we derive some arithmetic identities in Fq[T ] which may be nontrivial andof some interest.

Corollary 8.3. — Let q be a prime > 2 and d ≥ 1.1.

[d] = (−1)d+1 ∑P∈Ad+

(− Pqd−1

+d−1∑i=0

Pqd−1−(q−1)qi

)σP .

2. For 0 ≤ i ≤ d− 1,

(−1)d[i] =∑

P∈Ad+

Pqd−1−(q−1)qi

σP .

3.

(−1)dd∑i=1

[i] =∑

P∈Ad+

Pqd−1

σP .

Proof. — The first one follows from (17) and (19). For the second one, we first apply(16) to c = (q − 1)qi with 0 ≤ i ≤ d− 1 and get

(−1)dbqi(h) =∑

P∈Ad+

Pqd+1−1−(q−1)qi

σP =

∑P∈Ad+

Pqd−1−(q−1)qi

σP

where the last equality follows from qd+1−1−(q−1)qi = (q−1)qd+∑d−1j=0(q−1)qj−(q−1)qi

and degP = d. With (19), we get the second claim. The third one is obtained bycombining the first two identities.

26 C. ARMANA

Table 1. q = 3, d ≤ 4

i bi(h)1 02 −13 [1]5 −[1]6 −[1]2 − 19 [2] = [1]3 + [1]

14 [1]4 − 115 [1]5 − [1]3 + [1]18 −[1]6 + [1]4 − [1]2 − 127 [3] = [1]9 + [1]3 + [1]41 −[1]13 + [1]9 − [1]7 − [1]42 −[1]14 + [1]12 − [1]10 − [1]8 − [1]2 − 145 [1]15 − [1]13 + [1]11 − [1]9 + [1]3 + [1]54 −[1]18 + [1]12 + [1]10 − [1]6 + [1]4 − [1]2 − 181 [4] = [1]27 + [1]9 + [1]3 + [1]

In Table 1, we provide further examples of coefficients of h from Proposition 8.1.Observe that when i is even (resp. odd), bi(h) is an even (resp. odd) polynomial in[1] = T q − T . This is more generally true for any coefficient when q = 3: it follows fromthe coefficients of h being balanced, a property established by Gekeler (Theorem 2.4 of[9]). Note that, in our table, the constant term is −1 when i is even: we wonder if sucha statement holds more generally.

Acknowledgements

I am greatly indebted to Universität des Saarlandes (Germany) and Centre de RecercaMatemàtica (Spain), where this paper was written, for their pleasant hospitality. I wishto thank E.-U. Gekeler, L. Merel and D. Thakur for helpful comments on an earlierversion of the manuscript and D. Goss for his interest in this work.

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May 14, 2010 – revision: February 1, 2011C. Armana, Centre de Recerca Matemàtica – Campus de Bellaterra, Edifici C – E-08193 Bellaterra

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COEFFICIENTS OF DRINFELD MODULAR FORMS AND HECKE …armana.perso.math.cnrs.fr/armana-coefficientsdmfhecke-jnt.pdf · COEFFICIENTS OF DRINFELD MODULAR FORMS AND HECKE OPERATORS 3 Moreover,b